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Law of Cosines

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Cosine Rule

In trigonometry, the law of cosines formula which is also known as cosine rule formula or al-Kashi's theorem has relation with the lengths of each side of a triangle to that of the cosine of one of its angles. It is mostly useful for solving for missing information like an angle or a side in a triangle. For example, if all three sides of the triangle are known, the cosine rule helps one to find the measurement of any of the angles. In the same way, if two sides and the angle between them is known, the cosine rule can be used to find the length of the third side. 

 

Law of Cosines Problems and Solutions

Suppose we consider a triangle ABC, then according to the definition of cosine law, it can be stated that:

a2 = b2 + c2 – 2bc cos α

where a, b, and c are the measure of the sides of the triangle and α is the angle between sides b and c. 

Likewise, if β and γ are the angles between the sides ca and ab correspondingly, then according to the cosine law of triangle, we can also say that:

b2 = a2 + c2 – 2ac cos β

c2 = b2 + a2 – 2ab cos γ

The  Pythagorean theorem is generalized by the formula for the law of cosines, which holds true only for right-angled triangles. In this case, if either of the angles α, β or γ is a right angle i.e. having a measure of 90 degrees or 2 radians, as cos 90= 0, the law of cosine equation reduces to the Pythagoras’s theorem:

a2 = b2 + c2    [when α = 90 degrees]

b2 = a2 + c2    [when β = 90 degrees]

c2 = b2 + a2    [when γ = 90 degrees]

The law of cosine equation is useful for evaluating the third side of a triangle when the two other sides and their enclosed angle are known. When these angles are to be calculated, all three sides of the triangle should be known.

 

Proof of Law of Cosine Equation

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In the right triangle BAD, by the definition of cosine rule for angle :

cos A = AD/c

or

AD= c cosA

Subtracting above equation from side b, we get

DC = b – c cosA ……(1)

In the triangle BAD, according to the law of sine

sin A = BD/c

or

BD = c sinA ……(2)

In the triangle CDB, if we apply the Pythagorean Theorem, then

a2 = BD2 + DC2

Substituting for BD and DC from equations (1) and (2)

a2 = (c sin A)2 + (b – c cosA)2

On solving the above equation we get:

a2 = c2 sin2A + b2 – 2bccosA + c2 cos2A

Rearranging the above equation:

a2 = c2 sin2A + c2 cos2A + b2 – 2bc cosA

Taking out aas a common factor, we get;

a2 = c2(sin2A + cos2A) + b2 – 2bc cosA ………………. (3)

Now we know that,

sin2θ + cos2θ = 1

The equation (3) becomes:

∴ a2 = c2 + b2 – 2bc cosA

Hence, the law of cosines for angles is proved.


Application of Law of Cosines

1. To Find All the Angles of a Triangle Whose All Three Sides are Known:

In order to find the angles of a triangle ABC whose known measure of the three sides are a, b and c respectively, the law of cosines to find angle can be modified into the following:

cos α  = \[\frac{b^{2}+c^{2}-a^{2}}{2bc}\]  

cos β = \[\frac{a^{2}+c^{2}-b^{2}}{2ac}\]  

cos γ = \[\frac{a^{2}+b^{2}-c^{2}}{2ab}\]  

 

2. To Find the Lengths of the Sides of the Triangle When Two Other the Sides and the Angle Between them are Known:

To find the length of each side of triangle ABC, the formulae go as:

a = \[\sqrt{b^{2} + c^{2}-2bc cos \alpha}\]

b = \[\sqrt{a^{2} + c^{2}-2ac cos \beta}\]

c = \[\sqrt{b^{2} + a^{2}-2ab cos \gamma}\]

 

Did You Know?

Law of Cosines SSS 

In the law of cosine problems, when there is a case of SIDE SIDE SIDE congruency (SSS) between two triangles, all we need to find is the measure of the unknown triangle. Hence, by using the law of cosines, we can find the missing angle. 

Suppose, we find one angle using the law:

cos α  = \[\frac{b^{2}+c^{2}-a^{2}}{2bc}\]  

After that we can find the second angle again using the same law equation:

cos β = \[\frac{a^{2}+c^{2}-b^{2}}{2ac}\]  

Now by using the angle sum property of a triangle, we can simply find the third angle of the triangle. We need to subtract the sum of the two known angles from 180⁰ to get the accurate value of the third angle.

FAQ (Frequently Asked Questions)

Q1. When to Use the Cosine Law of Triangles? 

Answer: The law of sines and cosines is quite useful for finding:

  • the 3rd side of a triangle when we are aware of two sides and the angle between them

  • the angles of a triangle when we are familiar with all the 3 sides

Q2. What are the Easier Versions for Angles? 

Answer: We know how to find an angle when knowing 3 sides. It took a few steps, so it is easier to apply the "direct" cosine formula for triangle (which is just a reorder of the c² = a² + b² − 2ab cos(C) formula). It can also be in either of these forms:

cos(C) =  a² + b² − c² / 2ab

cos(A) =  b² + c² – a² / 2bc

cos(B) =  c² + a² − b² / 2ca

Q3. How Can You Remember the Law of Cosines Formula?

Answer: Well, it helps to understand it's the Pythagoras Theorem with a little extra so it works for all triangles:


Pythagoras Theorem:

(Just for Right-Angled Triangles)— a² + b² + c²


Law of Cosines:

(For all triangles)— a² + b² − 2ab cos(C) = c²

So, in order to remember it:

Imagine, "abc": a² + b² = c²,

Then a 2nd "abc": 2ab cos(C),

And put them collectively: a² + b² − 2ab cos(C) = c²

Q4. What are the Other Laws We Can Derive Using the Cosine Rule?

Answer: Following are the laws:-

  • Pythagorean Theorem

  • Triangle Inequality